Periodic solutions of nonlinear wave equations with general nonlinearities

Details Periodic solutions of nonlinear wave equations with general nonlinearities Periodic solutions of nonlinear wave equations with general nonlinearities

2002-11-20

arxiv.org/abs/math/0211310v1

Mathematics

Analysis of PDEs

29 pages

Scientific paper

10.1007/s00220-003-0972-8

We prove the existence of small amplitude periodic solutions, with strongly irrational frequency $ \om $ close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For $ \om $ close to one we prove the existence of a large number $ N_\om $ of $ 2 \pi \slash \om $-periodic in time solutions $ u_1, ..., u_n, ..., u_N $: $ N_\om \to + \infty $ as $ \om \to 1 $. The minimal period of the $n$-th solution $u_n $ is proved to be $2 \pi \slash n \om $. The proofs are based on a Lyapunov-Schmidt reduction and variational arguments.

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